A recent blog post on thatsmaths.com was about Tonenetz, a diagram of harmonic relationships of notes. Tonenetz means 'tone network' in German :-)
Readit Newsmci commented on The Tonnetz thetonnetz.com/... · Posted by u/mci
mci commented on An interactive guide to sensor fusion with quaternions quaternion.cafe/... · Posted by u/Bogdanp
> a quaternion is a 4D representation of a rotation in 3D space
A unit quaternion represents a rotation in R³.
A quaternion represents the quotient of two vectors in R³. That's what Hamilton had in mind.
mci commented on Cracovians: The Twisted Twins of Matrices marcinciura.wordpress.com... · Posted by u/mci
Agreed -- "is a term of the sum" is such an inverted way to look at it.
Better I think would be to say "the result in column i and row j is the sum of product of elements in column i of the left cracovian and column j of the right cracovian".
And even by this definition the example given doesn't seem to track (and the strangeness of sometimes saying "+" and sometimes not, and having both "0" and "-0" in the example is bananas!):
{ 3 2 } { 1 -4 } = { 5 -2 }
{ -1 0 } { -2 3 } = { 0 2 }
3 * 1 + -1 * -2 == 5 -- check
3 * -4 + -1 * 3 == -15 -- what?
2 * 1 + 0 * -2 == 2 (okay, but shouldn't this be in the lower left, column 1 dotted with column 2?)
2 * -4 + 0 * 3 = -8 (now I'm really missing something)I took the liberty to replace my awkward wording with your "the result in column i and row j is the sum of product of elements in column i of the left cracovian and column j of the right cracovian". Hope you don't mind. Thanks!
mci commented on Cracovians: The Twisted Twins of Matrices marcinciura.wordpress.com... · Posted by u/mci
I guess I'm skeptical of using a non-associative algebra instead of something that can trivially be made into a ring or field (i.e. matrix algebra). What advantages does this give us?
Author here. There are no practical advantages, as far as I know. Not even faster multiplication on today's computers.
mci commented on Cracovians: The Twisted Twins of Matrices marcinciura.wordpress.com... · Posted by u/mci
Agreed -- "is a term of the sum" is such an inverted way to look at it.
Better I think would be to say "the result in column i and row j is the sum of product of elements in column i of the left cracovian and column j of the right cracovian".
And even by this definition the example given doesn't seem to track (and the strangeness of sometimes saying "+" and sometimes not, and having both "0" and "-0" in the example is bananas!):
{ 3 2 } { 1 -4 } = { 5 -2 }
{ -1 0 } { -2 3 } = { 0 2 }
3 * 1 + -1 * -2 == 5 -- check
3 * -4 + -1 * 3 == -15 -- what?
2 * 1 + 0 * -2 == 2 (okay, but shouldn't this be in the lower left, column 1 dotted with column 2?)
2 * -4 + 0 * 3 = -8 (now I'm really missing something)Thanks for the feedback, everyone. I pasted my Polish text into Gemini to translate it into English. Gemini hallucinated the translation of this example. Now it should be OK.